Ultimately, whether or not the bank contracts with a third party vendor, the bank itself is responsible for making the flood determination. Back in the day, as my daughter calls it, we all used flood maps and did our own determinations. So the answer to your first question of whether the determination completed by the bank itself is acceptable is "Yes" (provided, of course, that it is accurate). So the bigger question is "Which determination is accurate?" The one from the determination firm is not automatically deemed to be more accurate than the one from the bank. In fact, especially in cases where the flood line might cross a particular piece of property but not include the improved structure, I've witnesses flood determination firms making inaccurate calls - they tend to call the property in if any part of the property is in the flood zone. So it may be that your lender looked at the map and looked at the survey and determined more accurately that the property was not in the flood zone.
I would respectfully disagree with Richard in this case. I don't think the best course is to leave this alone. As Andy says, investigate further to determine which determination is accurate. If the one that puts the improvements in the flood zone is accurate, then you must require flood insurance. Whether you eat the cost of the flood insurance or pass it on will have to be a PR and customer relationship call. But remember, there is always the possibility that the bank-completed determination is the accurate one and you will not need to take any further action.
Jim Bedsole, CRCM, CBA, CFSA
Opinions expressed are my own, and do not necessarily reflect those of my employer.